
Theorems REF  and REF  show that the differentials in the
spectral sequence (REF ) can be computed in terms of cup
products with \(H_{2i+1}\) 's. The existence of \(x_{p+2i}^{(t)}\) 's and
\(x_{p+2is}^{([\frac{t}{s}])}\) 's in Theorems REF  and REF 
plays an essential role in proving Theorems REF  and
REF , respectively. Theorems REF  and REF 
give a description of the differentials at the level of
\(E_{2t+3}^{p,q}\)  for the spectral sequence (REF ), which was
ignored in the previous studies of the twisted de Rham cohomology in
[1]}, [2]}.

Note that Theorem REF  is not a corollary of Theorem
REF , and it can not be obtained from Theorem REF 
directly.


